Resource Hacker is an utility to view, modify, rename, add, delete and extract resources in 32-bit and 64-bit Windows executables and resource files (*.res). The program can modify program icons as they appear in the taskbar, Start Menu and Desktop, among many other things.

Figured it out. I did a full install, took the files from the install folder and put them in a new folder, then created a file in that folder named reshacker.ini. With that file in place, Resource Hacker runs portably, writing settings to it. Thanks to Ennovy for mentioning the ini file and giving me the idea to try that. If anyone else has this problem, try that.